According to our setting, \(\Lambda _{z}=\Sigma _{z}^{-1}= \lambda L\)  [1]}, where \(L=D-A\)  is the Laplacian of a neighborhood graph defined on the pixel grid, \(D\)  is the graph degree matrix, \(A\)  is a pixel neighborhood adjacency matrix. Therefore, \(log|\Sigma  _{z}|\)  is constant. Since \(\Sigma _{z|x,y}\)  is set to be diagonal, \(log|\Sigma _{z|x,y}|=tr \ log\Sigma _{z|x,y}\) . And \(tr(\Sigma  _{z}^{-1}\Sigma _{z|x,y})=tr(\lambda (D-A)\Sigma _{z|x,y})=tr(\lambda D\Sigma _{z|x,y})\) . So we can get
\(\begin{aligned}&\mathcal {KL}[q_{\psi  }(z|y;x)||p(z)]\\&=\frac{1}{2}[tr(\lambda D\Sigma _{z|x,y}-log\Sigma _{z|x,y})+\mu _{z|x,y}^{T}\Lambda _{z}\mu _{z|x,y}]\\&+ const.\end{aligned}\) 
